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%% subsection 4.3 \it Supersymmetric Examples [slac-pub-7111-0-0-4-3 in slac-pub-7111-0-0-4: slac-pub-7111-0-0-4-4]
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\subsection{\usemenu{slac-pub-7111::context::slac-pub-7111-0-0-4-3}{\it Supersymmetric Examples}}\label{subsection::slac-pub-7111-0-0-4-3}
\label{SusyExamplesSubsection}
As a simple example, consider the contribution of an $N=4$
supersymmetry multiplet to a four-gluon amplitude.
This amplitude is an ordinary gauge-theory amplitude but with a
particular matter content:
one gluon, four gluinos and six real scalars all in the
adjoint representation. As discussed in section~\docLink{slac-pub-7111-0-0-2.tcx}[SWISubsection]{2.4},
$A_{4;1}^{\SUSY}(1^\pm, 2^+, 3^+, 4^+) = 0$ so the first non-trivial
case to consider is $A_{4;1}^\neqfour(1^-, 2^-, 3^+, 4^+)$.
For the $s$-channel cut depicted in fig.~\docLink{slac-pub-7111-0-0-4.tcx}[FourPtCutFigure]{9},
only the gluon loop contributes; for fermion
or scalar loops the supersymmetry identities in \eqn{SusyIdentities}
guarantee that at least one of the two tree amplitudes vanish.
The necessary tree amplitudes are the four-gluon amplitudes
$$
\eqalign{
& A_4^\tree(-\ell_1^+, 1^-, 2^-, \ell_2^+) = i { \spa1.2^4 \over
\spa{-\ell_1}.1 \spa1.2 \spa2.{\ell_2} \spa{\ell_2}.{-\!\ell_1}}\,, \cr
%
& A_4^\tree(-\ell_2^-, 3^+, 4^+, \ell_1^-) = i {
\spa{-\ell_1}.{\ell_2}^4 \over \spa{-\ell_2}.3 \spa3.4 \spa4.{\ell_1}
\spa{\ell_1}.{-\!\ell_2}} \,.\cr }
\equn
$$
All other combinations of helicities of the intermediate lines cause
at least one of the tree amplitudes on either side of the cut to vanish.
(The outgoing-particle helicity convention means that the helicity
label for each intermediate line flips when crossing the cut.)
Cut-constructibility of supersymmetric amplitudes allows us to use the
four-dimensional tree amplitudes, so that
the cut in the $s$ channel, \eqn{TreeProductDef}, becomes
$$
\eqalign{
A_{4;1}^\neqfour(1^-, 2^-, 3^+, 4^+)\Bigr|_{\scut}& =
\int {d^{4-2\eps} p \over (2\pi)^{4-2\eps}} \; {i \over \ell_1^2} \;
{i \spa1.2^4 \over \spa{\ell_1}.1 \spa1.2 \spa2.{\ell_2}
\spa{\ell_2}.{\ell_1}} \cr
& \hskip 20 mm \times \left.
{i\over \ell_2^2} \; {i\spa{\ell_1}.{\ell_2}^4
\over \spa{\ell_2}.3 \spa3.4 \spa4.{\ell_1} \spa{\ell_1}.{\ell_2}}
\right|_{s - \rm cut} \hskip -.6 cm \,, \cr}
\equn\label{SCutSusyA}
$$
where we have removed the minus signs from
inside the spinor products by cancelling constant phases. To put this
integral into a form more reminiscent of integrals encountered in
Feynman diagram calculations we may rationalize the denominators
using, for example,
$$
{1\over \spa2.{\ell_2}} = -{\spb2.{\ell_2} \over (p - k_1)^2} \, .
\equn\label{Rationalize}
$$
We use the on-shell conditions $\ell_1^2=0$ and
$\ell_2^2=0$, which apply even though the loop integral is
unrestricted, because of the $\scut$ restriction.
Performing such simplifications yields,
$$
\eqalign{
A_{4;1}^\neqfour&(1^-, 2^-, 3^+, 4^+)\Bigr|_{\scut} \cr
& \hskip -.3 cm
= - i A_4^\tree \!\left.\left[ \int\!
{d^{4-2\eps}p\over (2\pi)^{4-2\eps}} \;
{ {\cal N} \over p^2 (p - k_1)^4 (p - k_1 - k_2)^2 (p + k_4)^4}
\right]\right|_{\scut} \hskip - .6 cm \,, \cr}
\equn\label{SCutSusyB}
$$
where we have extracted a factor of the tree amplitude,
$$
A_{4}^\tree(1^-, 2^-, 3^+, 4^+) =
i {\spa1.2^4 \over \spa1.2 \spa2.3 \spa3.4 \spa4.1} \,,
\equn\label{ggggmmpptree}
$$
from the amplitude. The numerator of the integrand is
$$
\eqalign{
{\cal N} & = \spb{\ell_1}.1 \spa1.4 \spb4.{\ell_1} \spa{\ell_1}.{\ell_2}
\spb{\ell_2}.3 \spa3.2 \spb2.{\ell_2} \spa{\ell_2}.{\ell_1} \cr
& = \tr_+ [\ell_1 1 4 \ell_1 \ell_2 3 2 \ell_2] \cr
& = - 4 \tr_+[4321]\, \ell_1 \cdot k_1 \, \ell_1\cdot k_4
= -st \, (p - k_1)^2 (p + k_4)^2 \,, \cr}
\equn
$$
where $\tr_+[\cdots] = {1\over 2} \tr[(1+\gamma_5) \cdots]$
and we used
$$
\ell_1^2 = 0 \, ,
\hskip 1 cm \s\ell_1 \s\ell_2 = \s\ell_1 (\s k_3 + \s k_4) \, ,
\hskip 1 cm \s\ell_2 \s\ell_1 = -(\s k_1 + \s k_2) \s\ell_1 \, .
\equn
$$
The $\gamma_5$ term in the trace drops out because a four-point
amplitude has only three independent momenta to contract into the
totally anti-symmetric Levi-Civita tensor.
Thus in \eqn{SCutSusyB} the numerator neatly reduces the squared
propagators to single propagators,
$$
i s t A_4^\tree \int
{d^{4-2\eps}p\over (2\pi)^{4-2\eps}} \;
{1\over p^2 (p - k_1)^2 (p - k_1 - k_2)^2 (p + k_4)^2}
\, \biggr|_{\scut}\,,
\equn
$$
which is a scalar box integral. Thus the $s$-cut contribution is
given by
$$
A_{4;1}^\neqfour(1^-, 2^-, 3^+, 4^+)\Bigr|_{\scut}
= {- s t \over (4\pi)^{2-\eps}}\, A_4^\tree \, I_4(s,t)
\Bigr|_{\scut} \,,
\equn
$$
where the massless scalar box integral is
(see e.g. ref.~\cite{48})
$$
I_4(s,t) = - {2 \rg \over st} \biggl\{
- {1\over\e^2}\! \Bigl[ (-s)^{-\e}
+ (-t)^{-\e} \Bigr]
+ {1\over 2} \ln^2\left({s\over t}\right) + {\pi^2\over 2} \biggr\}
+ \Ord(\eps) \,.
\equn\label{BoxIntegral}
$$
The evaluation of the $t$-channel cut depicted in
\fig{FourPtCutFigure} is similar, but a bit more involved since all
particles in the multiplet contribute. However, after summing over
the contribution of all particles, with the help of the
SWI~(\docLink{slac-pub-7111-0-0-2.tcx}[SusyIdentities]{14}) and the Schouten
identity~(\docLink{slac-pub-7111-0-0-2.tcx}[SchoutenIdentity]{11}), the integral appearing in the
$t$-channel cut turns out to be the same as the one appearing in the
$s$-channel cut in \eqn{SCutSusyB}.
Combining the $s$ and $t$ channel results, the amplitude must be
$$
A_{4;1}^\neqfour(1^-,2^-,3^+,4^+)
= {- s t \over(4\pi)^{2-\eps}}\, A_4^\tree \, I_4(s,t) \, .
\equn\label{ggggmmppneqfour}
$$
The rational function proportional to $\pi^2$ contained in the box integral
(\docLink{slac-pub-7111-0-0-4.tcx}[BoxIntegral]{38}) is fixed by the cuts since it appears in
association with the logarithms in this function.
%
Integrals having cuts in multiple channels, such as
$I_4(s,t)$, provide a strong consistency check: their coefficients can
be obtained via two or more separate cut calculations and the results
must agree.
Following the same procedure one may evaluate the other nonvanishing
$N=4$ four-gluon amplitude, $A_{4;1}^\neqfour(1^-, 2^+, 3^-, 4^+)$,
where the negative helicities are non-adjacent. Surprisingly, the
same basic calculation can be easily extended to an arbitrary number
of external legs for maximally helicity violating (MHV) amplitudes,
those with two negative-helicity gluons and the remaining of positive
helicity. (A special case is $A_{5;1}^\neqfour(1^-,2^-,3^+,4^+,5^+)$,
given in~\eqn{gggggmmppploop}.) The cuts fall into two categories,
depending on whether the external negative-helicity gluons are on the
same or on opposite sides of the cut, as depicted in \fig{AllnFigure}.
In either case the tree amplitudes on both sides of the cuts are given
by the Parke-Taylor formula \cite{50,3},
$$
\eqalign{
A^\tree(\ell_1^+, m_1^+, & \ldots, k^-, \ldots, j^-,
\ldots, m_2^+,\ell_2^+) \cr
& = i { \spa{k}.j^4 \over
\spa{\ell_1}.{m_1} \spa{m_1,}.{m_1\!+\!1} \cdots
\spa{m_2\!-\!1,}.{m_2} \spa{m_2}.{\ell_2}
\spa{\ell_2}.{\ell_1}} \,,\cr}
\equn\label{PTAmplitudes}
$$
where $j$ and $k$ are the two negative-helicity legs, or by formul\ae\
related to \eqn{PTAmplitudes} by the SWI~(\docLink{slac-pub-7111-0-0-2.tcx}[SusyIdentities]{14}).
The key to evaluating the cut integrals
for an arbitrary number of external legs is that only two denominator
factors in the tree amplitudes~(\docLink{slac-pub-7111-0-0-4.tcx}[PTAmplitudes]{40}) contain the loop
momentum (since $1/\spa{\ell_2}.{\ell_1} =\spb{\ell_2}.{\ell_1}/(k_{m_1}
+ \cdots+ k_{m_2})^2$). Thus each tree contributes only two
propagators containing the loop momentum, so after including the two
cut propagators the hardest integral to be evaluated is the hexagon
integral depicted in \fig{HexagonFigure}. These hexagon integrals can
be reduced to scalar box integrals in much the same way as for the
four-point case, allowing one to obtain the amplitudes for an
arbitrary number of external legs \cite{14}.
%FIGURE
%
\begin{figure}
\begin{center}
\epsfig{file=Alln.eps,width=2.7in,clip=}
\end{center}
\vskip -.7 cm \caption[]{
\label{AllnFigure}
The relevant cuts for computing the MHV
amplitudes for an arbitrary number of external legs.}
\end{figure}
%
%FIGURE
%
\begin{figure}
\begin{center}
\epsfig{file=Hexagon.eps,width=1.1in,clip=}
\end{center}
\vskip -.7 cm \caption[]{
\label{HexagonFigure}
All-$n$ MHV supersymmetric amplitudes can be evaluated by evaluating
hexagon integrals.}
\end{figure}
%
%
The analysis of $N=1$ supersymmetric MHV amplitudes is similar,
although more complicated \cite{15}. Again the key to the
construction is that no more than six denominators contain loop
momentum, even for an arbitrary number of external legs. One instance
of the general $N=1$ MHV result is provided by
$A_{5;1}^\neqone(1^-,2^-,3^+,4^+,5^+)$ in~\eqn{gggggmmppploop}.
Notice that only the $s_{23}$ and $s_{51}$ channels contain cuts.
This result (which is also true for the scalar component) is a simple
consequence of the supersymmetry identities~(\docLink{slac-pub-7111-0-0-2.tcx}[SusyIdentities]{14}).
The construction of amplitudes via cuts does not rely on
supersymmetry, but only on the power-counting criterion; however,
non-supersymmetric amplitudes generally do not satisfy the criterion.